Transcript:
Hi, there now, let’s talk about the derivative of the greatest integer function recall the greatest integer function. It’s the function that basically rounds down so zero is equal to zero. But then when you get to one, you pop up to one and then when you go over to two, you pop up to two and over to three, you pop up to three over two for you pop up to four over to five and so on and down here, let’s see if we’re between Negative 1 and zero. We’re going to be down here, and it seems weak to zero pop up. So this is the greatest integer function that I’m plotting right now. Looks like so there we go. What does the derivative of this function look like the derivative of the greatest integer function? Well, it’s kind of hard to think about first Ik functions clearly growing, but look most the time it’s horizontal, which means the slope of the tangent line is 0 because the line tangent to the curve is 0 but here there. I’m not sure what’s been going on so the greatest integer function. I’m just going to tell you right now, but we’ll investigate here in a second. The derivative looks like this. It has open circles at every integer and then it’s 0 everyplace else. This is what the derivative of the greatest integer function looks like. I wonder if we can use the limit definition of the derivative to help us out. Suppose X is not an integer. So now, if we take the derivative of the greatest integer function, this is going to equal the limit as H goes to 0 of the greatest integer less than or equal to X, plus H minus the greatest integer less than or equal to X all over H. However, If we look at the plot again, imagine if X is right here. H we’re taking the limit as H goes to X, and so at some point, it’s going to be close enough to X that when you take the floor of X Plus H, it’s going to equal the floor of X, so this is going to be the limit as H goes to 0 of 0 because when H gets close to X, the greatest integer less than or equal to X plus H is equal to the greatest integer less than or equal to X over H and since we’re taking the limit as H goes to 0 we’re assuming that H is not equal to 0 so this whole thing is equal to the limit as H goes to 0 of 0 which is just equal to 0 so if X is not an integer, the derivative of the greatest integer function is equal to 0 But what happens when X is an integer? Now suppose X is an integer we’re going to see what happens when we try to compute the limit. That’s the definition of the derivative, and we’re going to do the limit from the left initially, so I’m going to look at the limit as H goes to 0 from the left and we’re looking at the limit of this function X plus H, the floor of X Plus H minus the greatest integer less than or equal to X all over H. OK, so now what’s going on here? Well, if X is an integer, then this is just equal to X, but remember H is going to 0 from the left. So this is the integer that’s 1 less than X, and so this is going to equal the limit as H goes to 0 from the left of minus 1 over. H now what’s going on here? Well, if H is going to 0 from the left, that means H is negative, right, so if H is negative. This is a small, negative number, a negative number over a small, negative number and so this limit is equal to infinity. But what about the limit from the right now again, suppose? X is an integer we’re going to look at the limit as H approaches zero from the right of the greatest senator, less than or equal to X, plus H minus the greatest integer less than or equal to X All over H well. If you look at the plot of the greatest integer function you can, you can see that this is equal to the limit as H goes to zero from the right of if it goes slightly to the right of X, It’s going to be the same value as X. So this is zero over H which is equal to zero. But what does this mean now? Let’s put our previous work together. So since when X is an integer, we have the limit from the left as H goes to zero equals infinity, and we have the limit from the right of as H goes to zero from the right of the greatest integer less than equal to X plus H minus Chris and your less than or equal to X is equal to zero. We see the left-hand limits and the right hand limits are not equal and so what this means is that the limit as H goes to zero doesn’t exist, which means that the derivative of the greatest integer less or equal to X does not exist, so we’ve seen that the derivative of the greatest integer function is zero, Except when X is an integer and what is an integer the derivative doesn’t exist. What what, what’s strange yet? Very simple function. Well, let’s do some more math you.